GACE Middle Grades Mathematics (4-8) (704) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rational number

A number expressible as a fraction a/b where a and b are integers and b ≠ 0; its decimal form terminates or repeats.

Irrational number

A real number that cannot be written as a ratio of integers; its decimal expansion is non-terminating and non-repeating, such as √2 or π.

Absolute value

The distance of a number from zero on the number line, always nonnegative; |x| = x if x ≥ 0 and |x| = −x if x < 0.

Scientific notation

A number written as a × 10^n where 1 ≤ |a| < 10 and n is an integer; useful for very large or very small quantities.

GCF and LCM

The greatest common factor is the largest integer dividing two numbers; the least common multiple is the smallest positive integer both divide. For positives, GCF × LCM equals their product.

Percent change

The ratio of the amount of change to the original amount, expressed as a percent: (new − old)/old × 100%.

Slope

The steepness of a line, computed as rise over run: m = (y₂ − y₁)/(x₂ − x₁). Parallel lines share the same slope.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a).

Discriminant

The expression b² − 4ac from the quadratic formula. Positive gives two real roots, zero gives one real root, negative gives two complex roots.

Vertex of a parabola

The highest or lowest point of y = ax² + bx + c, located at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Exponential function

A function of the form y = a · b^x with a ≠ 0 and b > 0, b ≠ 1; b > 1 models growth and 0 < b < 1 models decay.

Inverse function

A function f⁻¹ that reverses f, satisfying f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; its graph reflects f across the line y = x.

Arithmetic sequence

A sequence with a constant difference d between consecutive terms; the nth term is aₙ = a₁ + (n − 1)d.

System of linear equations

Two or more linear equations considered together; solutions are points satisfying all equations, found by substitution, elimination, or graphing.

Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c².

Area of a circle

The region enclosed by a circle equals π times the square of its radius: A = πr².

Volume of a cylinder

The space inside a cylinder equals the base area times height: V = πr²h, where r is the base radius and h the height.

Similar figures

Figures with equal corresponding angles and proportional corresponding sides; areas scale by the square of the scale factor and volumes by its cube.

Interior angle sum

The sum of the interior angles of a polygon with n sides is (n − 2) × 180°; for a triangle, n = 3 gives 180°.

Midpoint formula

The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Surface area of a sphere

The total area covering a sphere of radius r is S = 4πr²; its volume is V = (4/3)πr³.

Median

The middle value of an ordered data set; with an even number of values, it is the mean of the two middle values. It is resistant to outliers.

Interquartile range

The difference between the third quartile and first quartile, IQR = Q₃ − Q₁; it measures the spread of the middle half of the data.

Standard deviation

A measure of how spread out data are around the mean, roughly the typical distance of a value from the mean; larger values indicate greater variability.

Independent events

Events whose outcomes do not affect each other; for independent A and B, P(A and B) = P(A) × P(B).

Conditional probability

The probability of event A given that event B occurred: P(A|B) = P(A and B)/P(B), provided P(B) > 0.

Box plot

A display showing minimum, first quartile, median, third quartile, and maximum; the box spans Q₁ to Q₃ with a line at the median.

Correlation coefficient

A number r between −1 and 1 describing the strength and direction of a linear association; values near ±1 indicate strong linear relationships.

Mathematical modeling

Representing a real-world situation with mathematical structures such as equations or graphs, then interpreting and validating the results in context.

Mathematical justification

Supporting a claim with logical reasoning from definitions, properties, and established results, rather than relying only on examples.

Multiple representations

Describing the same mathematical relationship numerically, algebraically, graphically, and verbally; consistent translations among them support problem solving.

Error analysis

Examining a solution to locate the step where a mathematical rule was misapplied, explaining the error, and providing corrected reasoning.